| L(s) = 1 | − 2.71·3-s + 2.12·5-s + 1.38·7-s + 4.38·9-s − 6.32·11-s + 1.31·13-s − 5.76·15-s − 3.64·17-s + 3.20·19-s − 3.75·21-s − 5.96·23-s − 0.496·25-s − 3.76·27-s + 3.77·29-s + 3.53·31-s + 17.1·33-s + 2.93·35-s + 10.2·37-s − 3.57·39-s − 8.65·41-s + 7.34·43-s + 9.30·45-s − 12.6·47-s − 5.09·49-s + 9.90·51-s − 4.25·53-s − 13.4·55-s + ⋯ |
| L(s) = 1 | − 1.56·3-s + 0.949·5-s + 0.522·7-s + 1.46·9-s − 1.90·11-s + 0.364·13-s − 1.48·15-s − 0.884·17-s + 0.735·19-s − 0.819·21-s − 1.24·23-s − 0.0992·25-s − 0.725·27-s + 0.701·29-s + 0.634·31-s + 2.99·33-s + 0.495·35-s + 1.68·37-s − 0.571·39-s − 1.35·41-s + 1.11·43-s + 1.38·45-s − 1.85·47-s − 0.727·49-s + 1.38·51-s − 0.584·53-s − 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 + 6.32T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 - 3.20T + 19T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 8.65T + 41T^{2} \) |
| 43 | \( 1 - 7.34T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 4.25T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 + 9.73T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 5.74T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 0.796T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.540483247365180740799228846604, −8.236808689882641047810835286142, −7.56489611456286915607625481973, −6.29971451044108172997412952959, −5.96308708729689816632705041910, −5.05816839964983917846356789066, −4.56859288962152671624981869582, −2.76300011842685595138110106587, −1.57549360245531710690601916989, 0,
1.57549360245531710690601916989, 2.76300011842685595138110106587, 4.56859288962152671624981869582, 5.05816839964983917846356789066, 5.96308708729689816632705041910, 6.29971451044108172997412952959, 7.56489611456286915607625481973, 8.236808689882641047810835286142, 9.540483247365180740799228846604
